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Sum-Of-Squares Bounds via Boolean Function Analysis

2019
*
International Colloquium on Automata, Languages and Programming
*

We introduce a method for proving

doi:10.4230/lipics.icalp.2019.79
dblp:conf/icalp/Kurpisz19
fatcat:nwlhgdo6inav5cbj2fjacjuqti
*bounds*on the SoS rank based on*Boolean**Function**Analysis*and Approximation Theory. ... Finally, we consider the question by Bienstock regarding the instance*of*the Set Cover problem. For this problem we prove the SoS rank lower*bound**of*Ω( √ n). ... I C A L P 2 0 1 9 79:8*Sum*-*Of*-*Squares**Bounds**via**Boolean**Function**Analysis*Theorem 10. ...##
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Algorithm 959

2016
*
ACM Transactions on Mathematical Software
*

Moreover, because

doi:10.1145/2794077
fatcat:4tgey4sslzbxjfly4t3sra6hc4
*of*the size and complexity*of*modern ciphers, automatic*analysis*programs are very helpful in reducing the time required to study cryptographic properties*of*vector*Boolean**functions*. ... In addition, operations such as equality testing, composition, inversion,*sum*, direct*sum*, bricklayering (parallel application*of*vector*Boolean**functions*as employed in Rijndael cipher), and adding coordinate ... Search for Vector*Boolean**Functions*with Excellent Profiles*Boolean**functions*with very high nonlinearity pose some*of*the most challenging problems in the area*of*symmetric cryptography and combinatorics ...##
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Circuit and Decision Tree Complexity of Some Number Theoretic Problems

2001
*
Information and Computation
*

We extend the area

doi:10.1006/inco.2000.3017
fatcat:bqqmlfrysffl3fb6675w2tk4jy
*of*applications*of*the Abstract Harmonic*Analysis*to lower*bounds*on the circuit and decision tree complexity*of**Boolean**functions*related to some number theoretic problems. ... In particular, we prove that deciding if a given integer is*square*-free and testing co-primality*of*two integers by unbounded fan-in circuits*of**bounded*depth requires superpolynomial size. ... igor@mpce.mq.edu.au where the*sum*is taken over all strings w 2 B n*of*Hamming weight jwj > #. u t Note that a decision tree*of*size M can be simulated by the an unbounded fan-in*Boolean*circuit*of*...##
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Sum-of-squares hierarchies for binary polynomial optimization
[article]

2022
*
arXiv
*
pre-print

We consider the

arXiv:2011.04027v3
fatcat:io3h2ofz2bdktkh4ct3ma7ufqe
*sum*-*of*-*squares*hierarchy*of*approximations for the problem*of*minimizing a polynomial f over the*boolean*hypercube 𝔹^n={0,1}^n. ... This hierarchy provides for each integer r ∈ℕ a lower*bound*f_(r) on the minimum f_min*of*f, given by the largest scalar λ for which the polynomial f - λ is a*sum*-*of*-*squares*on 𝔹^n with degree at most ... Our objective in this paper is to investigate such an error*analysis*for this hierarchy applied to binary polynomial optimization as in (1) . 1.1. The*sum*-*of*-*squares*hierarchy on the*boolean*cube. ...##
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Optimal quantum query bounds for almost all Boolean functions

2013
*
Symposium on Theoretical Aspects of Computer Science
*

Our proof uses the fact that the acceptance probability

doi:10.4230/lipics.stacs.2013.446
dblp:conf/stacs/AmbainisBSW13
fatcat:ppztmbk3bnbplehbya7tf5ny7m
*of*a T -query algorithm can be written as the*sum**of**squares**of*degree-T polynomials. ... Optimal quantum query*bounds*for almost all*Boolean**functions*Ambainis, A.; Bačkurs, A.; Smotrovs, J.; de Wolf, R. ... Acknowledgement We thank Loïck Magnin and Jérémie Roland for sending us a copy*of*[16] . ...##
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Widening ROBDDs with Prime Implicants
[chapter]

2006
*
Lecture Notes in Computer Science
*

The number

doi:10.1007/11691372_7
fatcat:ijsr4atas5gwln4733t5admej4
*of*operations is, in turn, constrained by the number*of*times a*Boolean**function*can be weakened before stability is achieved. ... Despite the ubiquity*of*ROBDDs in program*analysis*, and extensive literature on ROBDD minimisation, there is a dearth*of*work on approximating ROBDDs. ... Previous attempts at*bounding*the iterations have confined the*analysis*to a fixed sub-domain*of**Boolean*formulae [13] . ...##
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Hilbert Function and Complexity Lower Bounds for Symmetric Boolean Functions

1999
*
Information and Computation
*

This paper explores the application

doi:10.1006/inco.1999.2798
fatcat:ot5k7c37wbazxlqd2y2uj6mzuu
*of*certain algebraic geometry techniques involving Hilbert*functions*and Gro bner bases to the*analysis**of*properties*of**Boolean**functions*. ... It gives some results and applications for symmetric*functions*. ] ... Similarly, in [1] , matrix*analysis*yields a lower*bound*for a coin weighing problem and, as a by-product, a generalization*of*a*bound*on weights for threshold gates. ...##
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An O(nlog log n) Learning Algorithm for DNF under the Uniform Distribution

1995
*
Journal of computer and system sciences (Print)
*

We show that a DNF with terms

doi:10.1006/jcss.1995.1043
fatcat:2xj62btz25bxzktokukrhhxusq
*of*size at most d can be approximated by a*function*with at most d O(d log1=") non zero Fourier coe cients such that the expected error*squared*, with respect to the uniform ... In this case our algorithm learns a DNF with a polynomial number*of*terms in time n O(loglog n) , and a DNF with terms*of*size at most O(log n= log log n) in polynomial time. ... Acknowledgements I would like to thank Eyal Kushilevitz for commenting on an early version*of*the paper. ...##
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Polynomial-Time Probabilistic Reasoning with Partial Observations via Implicit Learning in Probability Logics

2019
*
PROCEEDINGS OF THE THIRTIETH AAAI CONFERENCE ON ARTIFICIAL INTELLIGENCE AND THE TWENTY-EIGHTH INNOVATIVE APPLICATIONS OF ARTIFICIAL INTELLIGENCE CONFERENCE
*

In this work we consider the use

doi:10.1609/aaai.v33i01.33017866
fatcat:qfdy6vwqrzdhdcyhs3y5szchvq
*of**bounded*-degree fragments*of*the "*sum*-*of*-*squares*" logic as a probability logic. ... It is known that this logic is capable*of*deriving many*bounds*that are useful in probabilistic*analysis*. We show here that it furthermore captures key polynomial-time fragments*of*resolution. ... We thus observe that*sum*-*of*-*squares*is quite powerful for probabilistic*analysis*. We now turn to consider the*Boolean*reasoning ability*of**sum*-*of*-*squares*. ...##
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Computing the moments of k-bounded pseudo-Boolean functions over Hamming spheres of arbitrary radius in polynomial time

2012
*
Theoretical Computer Science
*

We show that given a k-

doi:10.1016/j.tcs.2011.02.006
fatcat:x2dzaivawzb2lee3k4wvjauhlu
*bounded*pseudo-*Boolean**function*f , one can always compute the cth moment*of*f over regions*of*arbitrary radius in Hamming space in polynomial time using algebraic information from ...*of*the region is exponential in the problem size. ... The authors would like to thank Schloß Dagstuhl -Leibniz Center for Informatics where some*of*the ideas contained in this paper were developed during the Dagstuhl seminar on the Theory*of*Evolutionary ...##
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Book announcements

1990
*
Discrete Applied Mathematics
*

A characterization

doi:10.1016/0166-218x(90)90013-3
fatcat:5jdut3iylrffvcnyiogm2tfe24
*of*a cone*of*pseudo-*Boolean**functions**via*supermodularity-type inequalities (Y. Crama, P.L. Hammer and R. Holzman). ... Contrasts, treatment comparisons and component*sums**of**squares*. Least*squares*estimators for linear models. Properties*of*least*squares*estimators. Overparameterisation and constraints. ...##
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Decision procedure based discovery of rare behaviors in Stochastic Differential Equation models of biological systems

2012
*
2012 IEEE 2nd International Conference on Computational Advances in Bio and medical Sciences (ICCABS)
*

Unfortunately, due to the limited availability

doi:10.1109/iccabs.2012.6182635
dblp:conf/iccabs/GhoshHJLJ12
fatcat:2evjoriuovf4rdth5qdkbqomdq
*of*analytic methods for SDEs, stochastic simulations are the most common means for estimating (or*bounding*) the probability*of*rare behaviors. ... We also compute the probability*of*an observed behavior under the assumption*of*Gaussian noise. ... Lemma 4: Given a lower*bound*on the probability density P*of*a discretized SDE behavior with m samples every ∆ time apart, the*sum**of**squares**of*increments*of*Brownian motion T ≡ m i=1 W ti − W ti−1 2 ...##
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Quantum boolean functions
[article]

2010
*
arXiv
*
pre-print

In this paper we introduce the study

arXiv:0810.2435v5
fatcat:suipqescfvhilnqvpy3n44dszy
*of*quantum*boolean**functions*, which are unitary operators f whose*square*is the identity: f^2 = I. ... Fourier coefficients*of**boolean**functions*; and two quantum versions*of*a theorem*of*Friedgut, Kalai and Naor on the Fourier spectra*of**boolean**functions*. ... TJO was supported by the University*of*London central research fund. ...##
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Polynomial-time probabilistic reasoning with partial observations via implicit learning in probability logics
[article]

2018
*
arXiv
*
pre-print

In this work we consider the use

arXiv:1806.11204v1
fatcat:4yf5xyqnzbesri4i7bfwkwbzhi
*of**bounded*-degree fragments*of*the "*sum*-*of*-*squares*" logic as a probability logic. ... It is known that this logic is capable*of*deriving many*bounds*that are useful in probabilistic*analysis*. We show here that it furthermore captures useful polynomial-time fragments*of*resolution. ... For*Boolean*reasoning,*sum*-*of*-*squares*simulates a variety*of*fragments*of*systems for which polynomial-time algorithms were already known. ...##
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A Brief Introduction to Fourier Analysis on the Boolean Cube

2008
*
Theory of Computing
*

We give a brief introduction to the basic notions

doi:10.4086/toc.gs.2008.001
dblp:journals/toc/Wolf08
fatcat:ijinjzubuzc4plzacz64xnn7na
*of*Fourier*analysis*on the*Boolean*cube, illustrated and motivated by a number*of*applications to theoretical computer science. ... Several more extensive surveys on Fourier*analysis**of**Boolean**functions*exist in the literature. ... Applying the KKL Inequality, for any δ ∈ [0, 1] we can*bound*the*sum**of**squared*biases by ∑ S∈( [n] k ) β 2 S = 2 2n |A| 2 ∑ S∈( [n] k ) f (S) 2 ≤ 2 2n δ k |A| 2 |A| 2 n 2/(1+δ ) ≤ 1 δ k 2 n |A| 2δ . ...
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